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Bordered Heegaard-Floer homology, category O, and higher representation theory

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Bordered Heegaard-Floer homology, category O, and higher representation theory
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12
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The Alexander polynomial for knots and links can be interpreted as a quantum knot invariant associated with the quantum group of the Lie superalgebra \mathfrak{gl}(1|1). This polynomial has been famously categorified to a link homology theory, knot Floer homology, defined within the theory of Heegaard-Floer homology. Andy Manion showed that the Ozsvath-Szabo algebras used to efficiently compute knot Floer homology categorify certain tensor products of \mathfrak{gl}(1|1) representations. For representation theorists, the work of Sartori provides a different categorification of these same tensors products using subquotients of BGG category \mathcal{O}. In this talk we will explain joint work with Andy Manion establishing a direct relationship between these two constructions. Given the radically different nature of these two constructions, transporting ideas between them provides a new perspective and allows for new results that would not have been apparent otherwise.